> Why is this better than just signing it with any private key and sharing the public key?
People using Ethereum often also use blockchain based domains which makes it easier for non-technical people to be more certain about endorsements. Ethereum leaves the implementation of how something becomes endorsed by a domain up to the owner(s) of a domain.
> Why do you need to involve a blockchain and I assume transaction fees as well?
This is more dependent on implementation. Just endorsing something does not need to be on chain, so there does not need to be fees.
People using Ethereum often also use blockchain based domains which makes it easier for non-technical people to be more certain about endorsements. Ethereum leaves the implementation of how something becomes endorsed by a domain up to the owner(s) of a domain.
> Why do you need to involve a blockchain and I assume transaction fees as well?
This is more dependent on implementation. Just endorsing something does not need to be on chain, so there does not need to be fees.